【作者】 曾霞；

【导师】 何平；

【作者基本信息】 西南交通大学 ， 应用数学， 2006， 硕士

【摘要】 随机变量之间的相依性是概率论与数理统计学中研究的最广泛的内容之一。但是传统的相依性指标对相依性的刻画有较大的局限性。近些年来利用copula刻画随机变量间相依性的理论越来越受到人们的关注。事实上copula不仅在相依性的研究中扮演着重要角色,还可以用来构造多元的分布函数族。如果我们有一族copula,那么根据Sklar定理,可以构造出任何指定边缘分布的二元或多元联合分布函数,而这些分布函数在建模与模拟方面是非常有用的。 有鉴于此,构造出多元的copula族就非常有意义。然而,多元copula的构造是非常困难的。尽管已有不少二元copula的构造方法,但是这些方法均难以扩展到多元的情形。现有文献中多元copula的构造方法也有较大的局限性,要求的前提条件亦相当苛刻。 本文借助于全概率公式的思想,提出了一种新的构造多元copula的方法。这种构造多元copula的方法是建立在本文定义过的g函数之上的。只要我们找到一族g函数,就可以求出对应的多元copula。本文首先介绍了copula的定义与基本性质,以及多元copula构造的研究现状,然后提出了构造二元copula的方法,并且将该方法扩展到多元copula构造的情形。同时讨论了通过该方法构造的二元copula的一些性质,而且多元copula的性质均可以方便的由相应二元copula的性质推广。由于这种多元copula的构造方法是基于g函数的,于是构造copula的关键是g函数的寻找。本文最后给出了一种基于一维分布函数来寻找g函数的方法,并且找到了在一定条件下,g函数与阿基米德copula之间生存元存在的一种联系。更多还原

【Abstract】 Dependence relations between random variables is one of the most widely studied subjects in probability and statistics. But the traditional correlation coefficients have some limits for a measure of the dependence between random variables. Using copula for measuring dependence takes more attentions in recent years. Actually, copula plays an important role in the study of dependence, it also can be used for constructing multivariate distributions. If there exist a family of copulas, we could construct any bivariate or multivariate distribution functions of given margins by Sklar’s theorem and these distributions are always very useful for simulation.Because this, it is meaningful of constructing multivariate copulas. But constructing n-copulas is difficult. Although there exist many ways about constructing bivariate copulas, few of them can extend to multivariate. Some methods about constructing n-copulas in many references have limits to some degree and also complicate premise.This paper gives a new method of constructing n-copulas depends on the law of total probability. The method that constructing n-copulas based on g functions which defined in this paper. We can get n-copulas if a family of g functions could be found. Firstly, this paper introduces the definition and basic properties of copula, and some results about construction of n-copulas in recent years. Secondly, the method of constructing 2-copulas was given, and then extends the construction procedures to n-copulas. And we have discussed some properties of the 2-copulas which construct by this method. Meanwhile all properties of 2-copulas in this paper have n-dimensional analogs. Since the method depends on g functions, finding g functions is the key to constructing n-copulas. Finally this paper have given a method of finding g functions depends on one-dimensional distributions. And the relations between gfunctions and the generators of Archimedean copula were found on someconditions.更多还原